Auslander–Reiten conjecture and Auslander–Reiten duality

Motivated by a result of Araya, we extend the Auslander–Reiten duality theorem to Cohen–Macaulay local rings. We also study the Auslander–Reiten conjecture, which is rooted in Nakayamaʼs work on finite dimensional algebras. One of our results detects a certain condition that forces the conjecture to hold over local rings of positive depth.